C of x equals 78,400 plus the 800x
plus x squared where c of x is the total cost
and x is the production level.
A, we want to find the production level
that will minimize the average cost and also b,
find the minimum average cost.
We need to be careful here because we're asked
to minimize the average cost, not the total cost.
So for the first half we want to find
the average cost function
where the average cost function is equal to
the total cost function divided by x,
where c of x represents the total cost
and x represents the production level.
So the average cost function
is equal to c of x,
divided by x or in our case,
78,400 plus 800x,
plus x squared,
divided by x.
So because we're dividing my a monomial,
we want to divide each term in the numerator by x.
So we'd have 78,400 divided by x,
plus 800x divided by x,
plus x squared divided by x.
So the average cost function
is equal to 78,400 divided by x,
plus this would just be 800,
plus this would just be x.
And now to minimize the average cost function,
our first step is to find the critical numbers
for this function which occurs where it's derivative
it equal to zero or undefined
or when the average cost and the marginal cost are equal.
We're going to use method a and determine when the derivative
of this function is equal to zero or undefined.
Notice here we have x to the first in the denominator
so if we move this up to the numerator
it would change the sign of the exponent.
So we can also write the average cost function
as 78,400x to the power of negative one,
plus 800,
plus x.
Now we'll find the derivative function on the next slide.
So the derivative of the average cost function
would be equal to the derivative of the first term.
We're going to multiply by negative one
so we have negative 78,400
and then subtract one from the exponent,
that'd be x to the power of negative two.
The derivative of 800 would be zero.
The derivative of x would be one.
So we have plus one.
So if we want to we can write this as
negative 78,400 divided by x squared.
If we move this back to the denominator,
it'll change the sign of the exponent.
And then we have plus one.
Notice how this would be undefined when x equals zero,
but when x equals zero, the production level is zero
and the average cost function is undefined.
So we'll set this equal to zero and solve for x
to determine the critical numbers.
Let's go ahead and add this fraction
to both sides of the equation.
That would give us one equals 78,400 divided by x squared.
And now here we can multiply both sides of the equation
by x squared or make this a proportion by putting this
over one and cross multiplying.
Notice one times x squared must equal one times 78,400.
So here we have x squared equals 78,400
and now we'll square root both sides of the equation,
but again, since x represents the production level,
we know x can't be negative
so we're only concerned about the principal square root
or the positive square root.
So now we'll go to the calculator.
We have the square root of 78,400.
So x equals 280.
So our only critical number is x equals 280.
So this should be the production level
that will minimize the average cost,
but to be sure, we should use either the first
or second derivative test to make sure the function
is minimized rather than maximized at this x value.
Let's go and use the second derivative test
to determine if the function is concave up or down
at x equals 280.
So here's the fist derivative.
So the second derivative would be equal to,
so to find the derivative of the first term,
we're going to multiply it by negative two
that's going to give us 156,800x
to the power of negative three.
The derivative of one would be zero.
Let's go ahead and write this as
156,800 divided by x cubed.
So notice how when x is equal to 280,
this would be positive or greater than zero,
which means the function would be concave up,
which is good news because that means at
this critical number we have a low point
or a minimum function value.
So to answer part a, a production level of 280
will minimize the average cost.
And now for part b, to find the minimum average cost,
we'll evaluate the average cost function here
when x equals 280.
Let's go ahead and write the average cost function
back in the form of 78,400,
divided by x,
plus 800,
plus x.
So the average cost, when x equals 280
would be 78,400,
divided by 280,
plus 800,
plus 280.
Now we'll go back to the calculator.
Notice how this sum here would be 1,080.
So the minimum average cost is $1,360.
I hope you found this helpful.